1. Problem-solving skills In most problems, you are given information about two points in space-time, and you are asked to find information about the space or time separation in another frame of reference. The single most universal way to attack such problems is to write down the 4-vector points in the one frame, and then transform them to the other frame to obtain the desired information.

2. Example: problem 5.14 For an observer in a rest frame S, an explosion occurs at x 1 =0, t 1 =0. A second explosion occurs at x 2 =500 m at time t 2 =10 -6 s. Calculate the velocity of a second observer if the second observer is to observe the two flashes simultaneously. Simultaneous: in the second observer’s frame, Write down the 4-vectors in frame S:

3. Construct the Lorentz transform to the other frame: Check the sign of the second terms: look at how the origin of the frame S (x=0) moves as seen in the frame S’: x decreases, so the second terms need the - sign. (It depends upon the statement of the problem and which way you assume the relative velocity to point.)  t=0 for simultaneity.

4. Invariants revisited We showed that is a Lorentz invariant: it has the same value for a given measure of (where and when) in all frames of reference. We will show that the same is true for the “dot product” of any two 4-vectors

5. So the scalar product of any two 4-vectors is a Lorentz invariant!

6. Energy and momentum How to treat energy and momentum in special relativity? We must recover two cases: Classical limit: v<<c, kinetic energy Relativistic limit: v  c, no matter can travel faster than the c, no matter how much kinetic energy it has.

7. Construct 4-vectors of energy, momentum Einstein included two terms in the energy: the kinetic energy T associated with motion, and the rest energy mc 2 associated with mass. We can evaluate the invariant easily in the rest frame: p=0, T=0, so

8. Now we can evaluate the velocity dependence of momentum, energy, kinetic energy, by making a Lorentz transform from the rest frame. Rest frame: Boost to velocity  c in x direction:

9. Now let’s recover the classical results: Indeed, we can define a relativistic transformation of mass:

10. Acceleration of a spaceship Suppose we accelerate a 10 ton spaceship to v=.5c. How much impulse do we deliver, how much work do we do? The output of a GW power plant for a year is 3x10 16 J! This is comparable to the energy consumption of the entire Earth’s population for a year.

11. Suppose we keep accelerating? As we try to increase  further, gamma increases quadratically. The impulse and energy increase rapidly: It would take an infinite amount of energy to reach v=c!